Are there any bounds on residues of $1/\zeta$ in roots of $\zeta$ in critical strip, which may use RH, but do not use the conjecture on simplicity of roots or something similar? I did not find such resuts in Titchmarsh, but I could miss something. Thanks!
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$\begingroup$ Could you say what kind of application you have in mind? $\endgroup$– David HansenMay 24, 2010 at 16:26
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1$\begingroup$ Yes. I express some contour integral of type $f(t)/\zeta(t)$ ($f$ decreases exponentially as imaginary pat of $t$ grows), as sum of residues. Then I would like this sum to be absolutely convergent. $\endgroup$– Fedor PetrovMay 25, 2010 at 8:20
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This is actually a very difficult problem, and currently most results are highly conjectural. It essentially comes down to finding useful bounds on discrete moments of the Riemann zeta function of the form $$J_k(T) = \sum_{0 < \Im(\rho) < T}{|\zeta'(\rho)|^{2k}},$$ as one can then choose the correct value of $k$ and apply partial summation.
For positive $k$, recent results of Milinovich and Milinovich and Ng show under the Riemann Hypothesis that $$T(\log T)^{(k+1)^2} \ll J_k(T) \ll T(\log T)^{(k+1)^2 + O(1/\log \log \log T)};$$ and slightly stronger results are known for $k = 0,1,2$ (with the latter two being under the Riemann Hypothesis).
For negative $k$, the situation is much more difficult. A conjecture of Gonek-Hejhal suggests that for all $k > -3/2$, $$J_k(T) \asymp T(\log T)^{(k+1)^2}$$ and this has been refined significantly by Hughes, Keating, and O'Connell. But it seems out of reach to prove anything significantly useful in this area; the best result so far has been by Gonek, who proved that $J_{-1}(T) \gg T$ assuming the Riemann Hypothesis and the simplicity of the zeroes of $\zeta(s)$. But this isn't useful for most applications, where an upper bound is needed. I believe it is possible to show $J_{-1} \ll T^{2+\varepsilon}$ under the Riemann Hypothesis and the simplicity of the zeroes of $\zeta(s)$, though I don't have a reference for this. Also it is quite possible that this result also holds if we replace $|\zeta'(\rho)|^{-2}$ by $\left|\mathrm{Res}_{s = \rho} \zeta(\rho)^{-1}\right|^2$, though again I don't know of a reference for this.